Scattering theory for the radialH˙12-critical wave equation with a cubic convolution

Abstract

In this paper, we study the global well-posedness and scattering for the wave equation with a cubic convolution ∂ 2 t u − u = ±(|x| −3 * |u| 2 )u in dimensions d ≥ 4. We prove that if the radial solution u with life-span I obeys (u, u t , then u is global and scatters. By the strategy derived from concentration compactness, we show that the proof of the global well-posedness and scattering is reduced to disprove the existence of two scenarios: soliton-like solution and high to low frequency cascade. Making use of the No-waste Duhamel formula and double Duhamel trick, we deduce that these two scenarios enjoy the additional regularity by the bootstrap argument of [7]. This together with virial analysis implies the energy of such two scenarios is zero and so we get a contradiction.

Generalized boundary value problems for analytic functions with convolutions and its applications

“…Note that P±, C i , and D i are bounded operators, and so does h. Since F is a boundedly reversible operator and its index is zero, hence h and FhF −1 have the same as the Noethericity and the indexes. Similar to the discussion in Miao et al and Blocki, 13,14 we know that FhF −1 is a Noether operator if and only if v(t) has no zero-point and pole-point (that is, p(t)q(t) ≠ 0). Assume that this is fulfilled, then Ind FhF −1 = Ind h. Thus, it is not difficult to obtain the Noethericity and the index of (5.27).…”

“…Since is a boundedly reversible operator and its index is zero, hence h and have the same as the Noethericity and the indexes. Similar to the discussion in Miao et al and Blocki, we know that is a Noether operator if and only if v ( t ) has no zero‐point and pole‐point (that is, p ( t ) q ( t ) ≠ 0). Assume that this is fulfilled, then .…”

Initial boundary value problem for a class of wave equations of Hartree type

“…For the Cauchy problem of Equation (), there is a large volume of literature on the global well‐posedness, the scattering theory, the blowup, and the asymptotical behavior of the solutions (see, e.g., Refs. 3–19 and the references therein). Furthermore, Tatar 20 studied the interaction between a dissipative term and a source term of cubic convolution type for the wave equation in $$R^n$$ $$\begin{eqnarray} &&u_{tt}+mu+\mu u_t(|x|^{-\gamma }*u^2_t)=\Delta u+\lambda h(t)u(|x|^{-\gamma }*u^2), \nobreakspace in\nobreakspace (0, T)\times R^n.$$…”